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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,5,32]))
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gp:[g,chi] = znchar(Mod(379, 704))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("704.379");
| Modulus: | \(704\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(704\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{704}(3,\cdot)\)
\(\chi_{704}(27,\cdot)\)
\(\chi_{704}(59,\cdot)\)
\(\chi_{704}(75,\cdot)\)
\(\chi_{704}(91,\cdot)\)
\(\chi_{704}(115,\cdot)\)
\(\chi_{704}(147,\cdot)\)
\(\chi_{704}(163,\cdot)\)
\(\chi_{704}(179,\cdot)\)
\(\chi_{704}(203,\cdot)\)
\(\chi_{704}(235,\cdot)\)
\(\chi_{704}(251,\cdot)\)
\(\chi_{704}(267,\cdot)\)
\(\chi_{704}(291,\cdot)\)
\(\chi_{704}(323,\cdot)\)
\(\chi_{704}(339,\cdot)\)
\(\chi_{704}(355,\cdot)\)
\(\chi_{704}(379,\cdot)\)
\(\chi_{704}(411,\cdot)\)
\(\chi_{704}(427,\cdot)\)
\(\chi_{704}(443,\cdot)\)
\(\chi_{704}(467,\cdot)\)
\(\chi_{704}(499,\cdot)\)
\(\chi_{704}(515,\cdot)\)
\(\chi_{704}(531,\cdot)\)
\(\chi_{704}(555,\cdot)\)
\(\chi_{704}(587,\cdot)\)
\(\chi_{704}(603,\cdot)\)
\(\chi_{704}(619,\cdot)\)
\(\chi_{704}(643,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((639,133,321)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 704 }(379, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
